$\otimes_{\mathbf W,\mathrm{ch}}$-markovian and imprimitive properties of block ciphers

In this paper, we describe relations between $\otimes_{\mathbf W,\mathrm{ch}}$-markovian block ciphers and a wreath product. Let $X$ be an alphabet of plaintexts (ciphertexts) in iterated block ciphers, $(X,\otimes)$ be a regular abelian group, and $\mathbf W=\{W_0,\dots,W_{r-1}\}$ be a partition of $X$. In the case when $\mathbf W$ is the set of cosets of a subgroup of $(X,\otimes)$, we prove that $\otimes$-Markov block cipher is $\otimes_{\mathbf W,\mathrm{ch}}$-markovian iff $\mathbf W$ is an imprimitivity system of the group generated by round functions of the cipher. We show that there are $\otimes_{\mathbf W,\mathrm{ch}}$-markovian block ciphers where $\mathbf W$ is not a set of cosets. So, for the additive group $(V_n^+,\oplus)$ of the vector space $V_n$, we describe $\oplus_{\mathbf W,\mathrm{ch}}$-markovian classes of nonlinear and affine transformations for $\mathbf W$ being not a set of cosets. We show that the set of all affine $\oplus_{\mathbf W,\mathrm{ch}}$-markovian transformations on $V_n$ is a group and give examples of it.